Painstakingly-Edited Typeset Lecture Notes

Physics 239: Topology from Physics Winter 2021 Lecturer: McGreevy These lecture notes live here. Please email corrections and questions to mcgreevy at physics dot ucsd dot edu. Last updated: May 26, 2021, 14:36:17 1 Contents 0.1 Introductory remarks............................3 0.2 Conventions.................................8 0.3 Sources....................................9 1 The toric code and homology 10 1.1 Cell complexes and homology....................... 21 1.2 p-form ZN toric code............................ 23 1.3 Some examples............................... 26 1.4 Higgsing, change of coefficients, exact sequences............. 32 1.5 Independence of cellulation......................... 36 1.6 Gapped boundaries and relative homology................ 39 1.7 Duality.................................... 42 2 Supersymmetric quantum mechanics and cohomology, index theory, Morse theory 49 2.1 Supersymmetric quantum mechanics................... 49 2.2 Differential forms consolidation...................... 61 2.3 Supersymmetric QM and Morse theory.................. 66 2.4 Global information from local information................ 74 2.5 Homology and cohomology......................... 77 2.6 Cech cohomology.............................. 80 2.7 Local reconstructability of quantum states................ 86 3 Quantum Double Model and Homotopy 89 3.1 Notions of `same'.............................. 89 3.2 Homotopy equivalence and cohomology.................. 90 3.3 Homotopy equivalence and homology................... 91 3.4 Morse theory and homotopy equivalence................. 93 3.5 Homotopy groups.............................. 95 3.6 The quantum double model........................ 103 3.7 Fiber bundles and covering maps..................... 108 3.8 Vector bundles and connections...................... 111 3.9 The quantum double model and the fundamental group......... 119 4 Topological gauge theories and knot invariants 124 4.1 Topological field theory........................... 124 4.2 Chern-Simons theory............................ 126 4.3 Links to the future............................. 137 2 0.1 Introductory remarks Goals. The primary, overt goal of this class is to use simple physical systems to introduce some important mathematical concepts, mostly in algebraic topology. For starters, this will include homology, cohomology and homotopy groups. When I say physical systems what I really mean is toy models of physical systems. The secondary, hidden goal is to smuggle in as much physics as possible. This will certainly include physics of topological phases of matter, about which there is a lot to say and a lot which is not yet understood. We will also talk about supersymmetry, a beautiful idea still looking for its rightful place in observable physics, but which has many deep connections to geometry and topology. So although the primary goal is mathematical, this is not a math course in many ways. One is that I will try to restrict myself to subjects where I think physical insight is helpful (or where I can at least find another good excuse). A brief overview of topology in many-body physics. There are many different manifestations of topology in physics, even just within condensed matter physics. Probably the manifestation of which the largest number of people are aware these days is band topology, or topological insulators. This is an example where the physics is extremely simple { it involves free fermions, so everything can be solved completely { but the mathematics is fancy (twisted equivariant K-theory). Maybe we'll get there. I am going to start instead with some situations where the physics is fancy or exotic { in the sense that it requires interactions or hasn't been found in earth-rocks yet { but the mathematics is stuff everyone should know: homology, cohomology, homotopy. I'm pretty excited that all of these things (which are squarely the subject of e.g. Hatcher's Algebraic Topology book) can be explained quite adequately using only familiar ideas from physics. In particular, all the (forbidding, homological) algebra of algebraic topology will take place in the comfort of a friendly Hilbert space. Generally covariant theories. There are many ways in which a physical system can be topological. One definition of topological is independent of a choice of metric (and therefore insensitive to distances between points). By this definition, historically the first topological theory then is actually general relativity. With general relativity (GR), we have a system defined on some smooth manifold without a choice of metric, because the metric is a fluctuating degree of freedom. In the language of path integrals (perhaps not entirely well-defined for GR), the metric is just an integration variable. (In low-enough dimensions this statement is known to be correct.) Towards the end of last century a large new collection of topological theories came 3 from several different directions. One is the study of solutions of equations which simply don't require a choice of metric. An example is Chern-Simons theory, with k R 2 action S[A] = 4π M tr A ^ F + 3 A ^ A ^ A : My point in writing the action here is to show that the metric does not appear. I hope I will say more later. Another, harder, example is the self-dual Yang-Mills equations { Donaldson theory of 4-manifolds. A more sophisticated origin of topological systems is supersymmetric field theories. Witten defined a procedure called twisting by which one can construct a set of observ- ables which do not depend on the metric. The fact that Donaldson theory also arises this way allows one to use the Seiberg-Witten solution of 4d N = 2 supersymmetric gauge theory to compute Donaldson invariants. I'll talk about first steps in this direction in x2. These topological field theories generally have the shortcoming that they are not unitary. What I mean by this is that they cannot arise as a low-energy description of a condensed matter system. Gapped phases of matter. This leads to a third origin of topological physics: gapped phases of quantum matter. First let's define the notion of a gapped quantum phase. A nice context is: consider a Hilbert space H = ⊗xHx made from finite-dimensional Hilbert spaces distributed over P space, and a local Hamiltonian H = x Hx. Local means Hx acts nontrivially only on degrees of freedom near x. Roughly, a groundstate has a gap if the energy difference ∆E to the first excited state stays finite in the thermodynamic limit (L ! 1, where L is the linear size of the system). E In contrast, a massless field in a box of linear size L has a level spacing of order 1=Lz, which vanishes in the thermodynamic limit. More precisely, we will allow some number of states below the gap, with a level spacing that decays faster than any power of L. (It is sim- ∆E plest when the number of such states is finite. But in fact in gapped fracton phases the number of these states diverges exponentially with L.) Most interesting will be the situation when any state obtained by groundstates acting on a groundstate by (superpositions of) local operators has a not related finite energy above the groundstates in the thermodynamic limit. If by local operators the putative groundstates were related by acting with a local operator, h 1j Ox j 2i= 6 0, we could add that operator to the Hamiltonian P ∆H = x cOx and split the degeneracy by a finite amount, so it would not be a stable situation. You may think this notion of having a gap is a property of the Hamiltonian and not just of the groundstate, but in fact a groundstate knows whether it is gapped or not. One signature is exponential decay of equal-time correlators of local operators. (I 4 don't know how to prove this; it is a piece of folklore.) Different gapped states are in different phases if we can't Wall of Gap-Closing space of H deform the Hamiltonian to get from one to the other B without closing the gap. The idea is that their ground- A states are related by adiabatic evolution. So it is tempt- ing to say that a gapped phase is an equivalence class of A0 Hamiltonians. In the figure at right, [A] = [A0] 6= [B]. Before you get too excited about the Wall of Gap-Closing: note that the closing of the gap does not by itself mean a quantum critical point: at a first order transition, just the lowest levels cross each other at some random point in parameter space. The two states which become degenerate are related by some horrible global rearrangement and not by acting with local operators, but the situation is unstable. So it's not necessarily true that any gapped state with a finite number of levels below the gap represents a phase of matter. A gapped state which does represent a phase of matter has an energy gap above a stable groundstate subspace. By stable I mean that there is an open set in the space of Hamiltonians in which the dimension of this subspace doesn't change. Actually, there is an important extra equivalence relation that we must include: We don't care if on top of some nontrivial phase of matter someone sprinkles a dust of decoupled qubits which are totally inert and do nothing at all. This represents the same phase of matter. Then, further, we are allowed to adiabatically deform the Hamiltonian1 including these decoupled bits, so that they can interact with the original degrees of freedom. So: in addition to allowing adiabatic variation of couplings, we also allow the addition of decoupled local degrees of freedom2. This definition is non-empty. An example of a gapped phase of quantum matter is obtained by putting a qubit on every site of some lattice, and taking H = H0 = P x − s Xs where Xs ≡ σs acts only on the qubit at site s as the Pauli x operator. Its groundstate is the product state ⊗s j!is, no matter what lattice we choose.

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